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{SECT 0 {PARA 256 "" 0 "" {TEXT -1 28 "Mindy Garner & Sherri Sisler" }
}{PARA 257 "" 0 "" {TEXT -1 14 "One HOT Party!" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 3 "" 0 "
" {TEXT -1 8 "Abstract" }}{PARA 0 "" 0 "" {TEXT -1 95 "We studied the \+
behavior of room temperature during a party, considering the thermosta
t setting." }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 10 "Background" }}
{PARA 0 "" 0 "" {TEXT -1 97 "This was something that our Applications \+
of Math class spoke about and we wanted to expand on it." }}}{SECT 1 
{PARA 3 "" 0 "" {TEXT -1 17 "Problem Statement" }}{PARA 0 "" 0 "" 
{TEXT -1 407 "     This problem involves a party that is to start at 8
 p.m. and end at 2 a.m.  Forty people are invited and are expected to \+
arrive during the first half-hour and to depart during the last half-h
our.  Given that that heat change int he house due to one person is ap
proximately .25 degrees F, we need to find the thermostat setting that
 would not allow the temperature of the house to rise above 77 degrees
." }}{PARA 0 "" 0 "" {TEXT -1 1106 "     A few assumptions were made i
n order to simplify the problem.  First of all, we assumed that we cou
ld use a given model of Newton's Law of Cooling, T ' = k*(M-T)+U+H, wh
ere M is the outside temperature, T is the inside temperature at any t
ime, U is an external negative change in temperature acting on the sys
tem, and H is an external positive change in the temperature acting on
 the system.  This equation models a house's temperature throughout an
 ordinary day, starting as 8 a.m.  Another assumption is that, up unti
l the beginning of the party, the thermostat is set on 70 degrees.  We
 also assumed that the first and last half-hours, when guests are arri
ving and departing, could be ignored.  The reasoning behind this assum
ption is that, in solving this problem, we are looking at the maximum \+
room temperature over the entire period of the party.  This maximum wi
ll not be reached until the total number of guests, forty, have arrive
d.  One last assumption was made; the thermostat is only accurate to o
ne decimal place.  In other words, one possible setting for the thermo
stat is 75.2 degrees. " }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 17 "Model \+
Formulation" }}{PARA 0 "" 0 "" {TEXT -1 595 "     We used k=3/16 for t
his situation.  This number was found by experimentation (and will cha
nge for different homes).  For the home the experiment was done in, k \+
was found as follows:  when M=60 and T=68, the heat was turned off and
 the change in temperature was observed to be a drop of 6 degrees in 4
 hours, the outside temperature remained about 60 degrees.  In this co
nstant outside temperature, no heat from a heater, no heat from other \+
sources, k must satisfy T ' = k*(M-T).  T ' is approximated by \"chang
e in T\" / \"change in t\", (t is time),  so: (68-62) / (-4) = k*(60-6
8) ... k=3/16." }}{PARA 0 "" 0 "" {TEXT -1 328 "     We used M = 21/2 \+
* sin (2*Pi*t / 24) + 141/2 for this situation.  This models the outsi
de temperature assuming that we have a high of 81 degrees at 2pm and a
 low of 60 degrees about 12 hours later (and t=0 is 8am).  The followi
ng plot shows the curve modeling the temperature as it changes from 8a
m one day to 8am the next." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
48 "plot(21/2 * sin (2*Pi*t / 24) + 141/2, t=0..24);" }{TEXT -1 0 "" }
}{PARA 13 "" 1 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 338 "     \+
We used U=k ' (thermostat-T) because the temperature inside varies acc
ording to Newton's Law also.  k ' is the thermostat changing constant \+
and was found by experimentation also.  When T=68 and the thermostat i
s set at 72 degrees, the temperature in the house raised 2.5 degrees i
n 15 minutes.  Therefore k ' = 5/2 in this situation." }}{PARA 0 "" 0 
"" {TEXT -1 141 "     We can easily see that H=40*.25, because each pe
rson raises the house temperature by .25 degrees, and there were 40 pe
ople at the party." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 108 "     Our final differential equation modeling the temper
ature inside the house during a party is as follows:" }}{PARA 0 "" 0 "
" {TEXT -1 91 "     T ' = 3/16 [ 21/2 sin (2*Pi*t / 24) + 141/2 - T ] \+
+ 5/2 [ thermostat - T ] + .25 * 40." }}{PARA 0 "" 0 "" {TEXT -1 0 "" 
}}{PARA 0 "" 0 "" {TEXT -1 296 "     In order to find the initial cond
ition, we looked at the above plot at t=12, which is 8 pm, the startin
g time of the party. We assumed that the outside temperature at t=12 i
s the same as the inside temperature.  By clicking on the plot at the \+
right time, we find that T(12) = 70.2 degrees.  " }}}{SECT 1 {PARA 3 "
" 0 "" {TEXT -1 7 "Results" }}{PARA 0 "" 0 "" {TEXT -1 341 "Now that w
e have our complete differential equation, we can choose some random v
alues for the thermostat setting and make a graph of what happens to t
he house temperature at those settings.  It is important to remember t
hat our objective is to find the maximum thermostat setting that will \+
not allow the room to get hotter than 77 degrees.  " }}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 14 "with(DEtools):" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 202 "DEplot(diff(T(t),t) = 3/16 *\n      (21/2*sin(2*Pi
*t/24) + 141/2 - T(t)) + 5/2 *\n      (70 - T(t)) + .25*40, T(t), t=12
..18,\n      [[T(12) = 70.2]], method = rkf45, stepsize = .1,\n       \+
arrows = none); " }}{PARA 13 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 203 "DEplot(diff(T(t),t) = 3/16 *\n      (21/2*s
in(2*Pi*t/24) + 141/2 - T(t)) + 5/2 *\n      (75 - T(t)) + .25*40, T(t
), t=12..18,\n      [[T(12) = 70.2]], method = rkf45, stepsize = .1,\n
       arrows = none); \n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
203 "DEplot(diff(T(t),t) = 3/16 *\n      (21/2*sin(2*Pi*t/24) + 141/2 \+
- T(t)) + 5/2 *\n      (73.7 - T(t)) + .25*40, T(t), t=12..18,\n      \+
[[T(12) = 70.2]], method = rkf45, stepsize = .1,\n       arrows = none
);" }{TEXT -1 0 "" }{MPLTEXT 1 0 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}}{PARA 0 "" 0 "" {TEXT -1 238 "     It is obvious that the tempera
ture gets fairly close to 77 degrees around 9:30 pm, but never reaches
 it.  This final graph suggests that the best setting for the thermost
at is 73.7 degrees, so that the house never reaches 77 degrees." }}}
{SECT 1 {PARA 3 "" 0 "" {TEXT -1 10 "Variations" }}{PARA 0 "" 0 "" 
{TEXT -1 306 "     The specific result that we obtained is not very us
eful to anyone else.  There will be different k's for different homes \+
and thermostats.  Also, on any given day, the temperature will vary fr
om our model due to weather conditions and the time of year because th
e sun rises and sets at different times." }}}{SECT 1 {PARA 3 "" 0 "" 
{TEXT -1 10 "References" }}{PARA 0 "" 0 "" {TEXT -1 100 "     Dr. Holl
y P. Hirst, Appalachian State University, Math Department, email: hph@
math.appstate.edu" }}}}{MARK "3 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 }